Foundations of Higher Mathematics
This text introduces students to basic techniques of writing proofs and acquaints them with some fundamental ideas. The authors assume that students using this text have already taken courses in which they developed the skill of using results and arguments that others have conceived. This text picks up where the others left off -- it develops the students' ability to think mathematically and to distinguish mathematical thinking from wishful thinking.
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The Logic and Language of Proofs
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abelian accumulation point assume Axiom of Choice belongs Cantor cardinal number Cauchy sequence chapter congruence classes consider continuous at x0 contrapositive converges countable set countably infinite counterexample denote directed graph divides divisible divisor domain equation equivalence classes equivalence relation exists Figure finite set following theorem formula given hence Hint homomorphism infinite set integer isomorphic Let G Let x e Mathematical Induction multiplication natural number nonempty set notation number greater number n odd integer one-to-one function one-to-one function mapping operation ordered pairs partial order partition permutations Pigeonhole Principle positive integers positive number positive real numbers prime number Principle of Mathematical problem PROOF Let PROOF See Exercise PROOF Suppose Prove by induction Prove Theorem Q is true rational numbers real number reflexive Schroeder-Bernstein Theorem Second Principle set theory subgroup of G symmetric tion uncountable set vertices write